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The Many Names of (7,3,1) and the Unity of Discrete Mathematics

Ezra “Bud” Brown fell hard for (7,3,1). Hard and fast. When he first learned about this mathematical object—in its guise as a symmetric balanced incomplete block design—in graduate school, “something just glowed inside,” Brown told a crowd at MAA’s Carriage House.

The Virginia Tech mathematician devoted his contribution to MAA’s Distinguished Lecture Series to “The Many Names of (7,3,1) and the Unity of Discrete Mathematics,” helping the audience understand what has proved to be an enduring fascination for him.

To introduce the object that he advertised as tying together everything from round-robin tournaments to error-correcting codes, Brown enlisted the first of many historical figures to populate his talk. He paraphrased a question English clergyman Thomas Kirkman posed in a magazine in the 1840s: Is it possible to arrange a set of seven items into seven three-element subsets—call them blocks—such that each item is in three blocks and each pair of items is in exactly one block together?

The answer, as Kirkman knew, is “yes,” and Brown led listeners through the construction of such an arrangement—the only such arrangement, it turns out, up to changing the names—using the set comprising the counting numbers 1 through 7. (We leave this as an exercise for the reader.) “This is the (7,3,1) block design,” Brown said, before going on to define a generalization of that idea.

When Brown first met the captivating block design, the man facilitating the encounter—David Roselle—told the young Bud about some “other disguises under which (7,3,1) showed up.” Brown has since run with the idea (perhaps further than his combinatorics professor ever dreamed), enumerating no fewer than 21 (7*3*1) alternative formulations for the object of his affection.

Take the Fano Plane, named—unfairly, Brown noted, since Kirkman thought of it earlier—for Italian geometer Gino Fano. It’s a projective plane with seven points and seven lines. Each line has three points, each point lies on three lines, and each pair of points defines a unique line. The seven three-point lines, in other words, are exactly the blocks of the (7,3,1) block design.

Brown outlined with particular relish the connection between Latin squares—familiar from Sudoku puzzles—and (7,3,1). It has to do with a property called orthogonality, and it afforded Brown the opportunity to recount how the preternaturally insightful Leonhard Euler was once “consummately, spectacularly wrong.”

Bud Brown is not the only one to get excited about this. When, in 1959, Ernest T. Parker constructed a pair of orthogonal Latin squares of order 10—something that Euler had conjectured impossible 175 years previously—the squares made the cover of Scientific American. It was, Brown enthused, a “singular event in the history of mathematics.”

Brown couldn’t savor Euler’s error for long, however. The (7,3,1) design manifests itself in so many and such marvelous ways throughout mathematics that Brown didn’t have time to dwell on the legend’s rare lapse—or on anything else. He sped through difference sets. He talked of incidence matrices and Hamming codes and showed how seven mutually adjacent hexagons can tile a torus.

Bud Brown answered questions from the crowd after his MAA Distinguished Lecture.

Before long, the names for (7,3,1) came so fast and furious that Brown hardly tried to explain them: skew-Hadamard matrix, Klein’s quartic curve, Leech’s eight-dimensional minimal sphere-packing lattice. And when he got to one particularly outlandish-sounding claim—that (7,3,1) can be viewed as “a quantum-controlled 2-junction NOT gate”—even Brown had to concede bewilderment. “I have no idea what that is,” he admitted with a laugh.